Related papers: Formalizing Termination Proofs under Polynomial Qu…
We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the…
We state a combinatorial optimization problem whose feasible solutions define both a decomposition and a node labeling of a given graph. This problem offers a common mathematical abstraction of seemingly unrelated computer vision tasks,…
To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…
We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer…
Motivated by a growing list of nontraditional statistical estimation problems of the piecewise kind, this paper provides a survey of known results supplemented with new results for the class of piecewise linear-quadratic programs. These are…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The…
The submodular partitioning problem asks to minimize, over all partitions $P$ of a ground set $V$, the sum of a given submodular function $f$ over the parts of $P$. The problem has seen considerable work in approximability, as it…
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all…
Programs with multiphase control-flow are programs where the execution passes through several (possibly implicit) phases. Proving termination of such programs (or inferring corresponding runtime bounds) is often challenging since it…
We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions…
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…
Graph partitioning is a key fundamental problem in the area of big graph computation. Previous works do not consider the practical requirements when optimizing the big data analysis in real applications. In this paper, motivated by…
The scope of this work is the constraint-based synthesis of termination arguments for the restricted class of programs called linear lasso programs. A termination argument consists of a ranking function as well as a set of supporting…
We carry out a proof theoretic analysis of the wellfoundedness of recursive path orders in an abstract setting. We outline a very general termination principle and extract from its wellfoundedness proof subrecursive bounds on the size of…
Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they…
Let $\operatorname{Con}(\mathbf T)\!\restriction\!x$ denote the finite consistency statement "there are no proofs of contradiction in $\mathbf T$ with $\leq x$ symbols". For a large class of natural theories $\mathbf T$, Pudl\'ak has shown…
Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof…
We introduce the concept of sos-convex Lyapunov functions for stability analysis of both linear and nonlinear difference inclusions (also known as discrete-time switched systems). These are polynomial Lyapunov functions that have an…
We present a new approach to termination analysis of logic programs. The essence of the approach is that we make use of general orderings (instead of level mappings), like it is done in transformational approaches to logic program…